The Direct Solution of the Discrete Poisson Equation on a Rectangle

Abstract

where G is a rectangle, Au = 82u/8x2 + 82u/8y2, and v, w are known functions. For computational purposes, this partial differential equation is frequently replaced by a finite difference analogue. These discrete models for (1) consist of linear systems of equations of very large dimension, and it is widely recognized that the usual direct methods (e.g., Gaussian elimination) are unsatisfactory for such systems [18, ?? 21.2-21.3]. Theoretical investigation has, therefore, been primarily directed toward the development of effective iterative methods for the solution of these problems [64], [66]. In recent years, however, direct methods that take advantage of the special block structure of these linear equations have appeared. For the rectangular regions under consideration, these methods can be considerably faster than iterative methods. The purpose of this survey paper is to provide brief summaries and a list of references for methods which can be used to directly solve the finite difference equations. Some of these methods can be applied to problems in more general domains. However, the extensions generally include only simple rectilinear regions, such as L-shaped or T-shaped domains. This is basically due to the fact that the direct methods require a great degree of regularity in the block structure of the matrix equation. In our discussion, we will indicate whether the methods are easily adaptable to more general regions, and to more general elliptic partial differential equations.